Question

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R. Assertion A: In a central force field, the work done is independent of the path chosen. Reason R: Every force encountered in mechanics does not have an associated potential energy. In the light of the above statements, choose the most appropriate answer from the options given below.

• A. A is true but R is false
• B. Both A and R are true and R is the correct explanation of A
• C. Both A and R are true but R is NOT the correct explanation of A
• D. A is false but R is true

Hint:

Distinguish between conservative and non-conservative forces when discussing work and energy: conservative forces have path-independent work and associated potential energy.

Answer 1

The Correct Option is C

Assertion A: In a central force field, the work done is independent of the path chosen. This statement is true.
Central forces are conservative forces, and the work done by a conservative force depends only on the initial and final positions, not on the path taken.

Reason R: Every force encountered in mechanics does not have an associated potential energy. This statement is also true. Non-conservative forces like friction and air resistance do not have associated potential energies because the work done by these forces depends on the path taken.

Analysis of the Relationship Between A and R:

• The independence of work done on the path depends on the fact that the force is conservative.
• All conservative forces possess an associated potential energy function.
• Since Reason R states that not every force has an associated potential energy, while Assertion A refers specifically to conservative (central) forces which do have potential energy,
• Reason R is not a correct explanation of Assertion A — it makes a general statement about forces, not specifically about central or conservative forces.

Therefore, both A and R are true, but R is NOT the correct explanation of A.

Final Answer:
The final answer is $ (2)\ \text{Both A and R are true but R is NOT the correct explanation of A} $.

Answer 2

Step 1: Understand the given statements.
Assertion (A): In a central force field, the work done is independent of the path chosen.
Reason (R): Every force encountered in mechanics does not have an associated potential energy.

Step 2: Analyze the assertion (A).
A central force field is one in which the force on a particle is directed toward (or away from) a fixed point and its magnitude depends only on the distance from that point. Examples include gravitational and electrostatic forces.
For central forces, the work done depends only on the initial and final positions, not on the path followed. Hence, the work done in a central force field is path independent.
✅ Assertion (A) is true.

Step 3: Analyze the reason (R).
The reason states that every force encountered in mechanics does not have an associated potential energy. This is true because not all forces are conservative. For example, friction and viscous forces are non-conservative, and hence no potential energy function can be defined for them.
✅ Reason (R) is also true.

Step 4: Check the explanation relationship between A and R.
Although both (A) and (R) are true statements, (R) does not correctly explain (A). The independence of work done from the path in a central force field arises because such forces are conservative, not because some forces lack potential energy.

Step 5: Conclusion.
Both (A) and (R) are true, but (R) is not the correct explanation of (A).

Final Answer:
Both A and R are true but R is NOT the correct explanation of A.